Computational complexity of small descriptions

Abstract

For a set L that is polynomial time reducible (in short, = sub T super P-reducible) to some sparse set, we investigate the computational complexity of such sparse sets relative to L. We construct sets A and B such that both of them are = sub T super P-reducible to some sparse set, but A (resp., B) is = sub T super P-reducible to no sparse set in P super A (resp., NP super B ¿ co-NP super B); that is, the complexity of sparse sets to which A (resp., B) is = sub T super P-reducible is more than P super A (resp., NP super B ¿ co-NP super B). Some consequences of these results and applications of our proof technique are also discussed.